Abstract

Let q be a prime. It is simple to show that SLn(Z[1/q]) is dense in SLn(R ), and we want to make this quantitative. Equivalently, we want to study the density of the q-Hecke orbit of a point on the locally symmetric space SLn(Z[1/q]) SLn(R )/SO(n). This problem was studied in great generality by Ghosh-Gorodnik-Nevo, who defined a ”Diophantine exponent” to measure the density of the orbit. A similar definition appears in the work of Sarnak and Parzanchevski on Golden Gates. Assuming the Generalized Ramanujan Conjetcure (GRC), we prove that the Diophantine exponents in our case are optimal. Unconditionally, we prove that the exponents are optimal for n=2 and n=3, and are almost optimal for general n. The proof combines ”density bounds” towards GRC by Blomer with new bounds on the L^2-growth of Eisenstein series in a compact domain which we develop, and are of independent interest. Based on ongoing work with Subhajit Jana.